This invention relates to shim coils. In particular, the invention relates to shim coils suitable for use in magnetic resonance applications that generate tesseral fields located asymmetrically in a finite-length coil. A method for the design of such shim coils of the type useful for Magnetic Resonance applications is described. The method involves a type of target-field approach, but the exact geometry of the shim coils is treated without approximation. In particular, the fact that shim coils are of finite length is catered for. Although illustrated herein in terms of shim coils, the methods of the invention can also be used to design essentially any type of coil which is to be used to produce a desired magnetic field, including, without limitation, gradient coils and H0-producing coils.
In magnetic resonance imaging (MRI) applications, a patient is placed in a strong and homogeneous static magnetic field, causing the otherwise randomly oriented magnetic moments of the protons, in water molecules within the body, to precess around the direction of the applied field. The part of the body in the homogeneous region of the magnet is then irradiated with radio-frequency (RF) energy, causing some of the protons to change their spin orientation. When the RF energy source is removed, the protons in the sample return to their original configuration, inducing a measurable signal in a receiver coil tuned to the frequency of precession. This is the magnetic resonance (MR) signal. Most importantly, the frequency at which protons absorb the RF signal depends on the background magnetic field.
In practice, the presence of the patient""s body perturbs the strong magnetic field slightly, and so shim coils are used to correct the field, to give the best possible final image. The field within a specified target volume (the diameter of the sensitive volume, or DSV) is typically represented in terms of spherical harmonics, and so impurities in the field are analyzed in terms of the coefficients of an expansion in these harmonics. Shim coils are therefore designed to correct a perturbed magnetic field by producing a particular spherical harmonic that can be added to the background magnetic field, so as to cancel the effect of a certain harmonic caused by an impurity. Many of these coils may be present in a particular MRI device, and each may have its own power supply, to produce the required current flow.
The major task associated with the design of these shim coils is to determine the precise windings on the coil that will produce the desired magnetic field within the coil. One method, due to Turner (1986, A target field approach to optimal coil design, J. Phys. D: Appl. Phys. 19, 147-151), is to specify a desired target field inside the cylinder, at some radius less than the coil radius. Fourier transform methods are then used to find the current density on the surface of the coil, required to give the desired target field. This method has been widely used and is successful in applications, but it is based on the approximation that the coil is, in some sense, infinite in length so that the Fourier transform technique can be applied. Finite-length coils can be simulated in this technique by adding a constraint that the current density must fall to zero outside some finite interval, and this is discussed by Turner in U.S. Pat. No. 5,289,151. Nevertheless, coils of finite length are not natural to this approach, and in some circumstances smoothing functions have to be incorporated in the Fourier transform so as to guarantee its convergence.
A related method for overcoming the difficulty associated with coils of finite length has been advanced by Forbes, Crozier and Doddrell in Australian Provisional Patent Application PQ9787 (see U.S. Pat. No. 6,377,148 B1) and Forbes and Crozier (2001, Asymmetric zonal shim coils for Magnetic Resonance applications, Med. Phys. 28, 1644-1651). The technique employs a target-field approach and builds in the finite length of the coils by making use of a Fourier series technique. This approach involves approximations, but is capable of designing coils for asymmetrically located fields in a very systematic way.
An alternative method for the design of coils of finite length is the stochastic optimization approach pioneered by Crozier and Doddrell (1993, Gradient-coil design by simulated annealing, J. Magn. Reson. A 103, 354-357). This approach seeks to produce a desired field in a given volume (the DSV) using optimization methods to adjust the location of certain loops of wire and the current flowing in those loops. The method is very robust, since it uses simulated annealing as its optimization strategy, and it can incorporate other constraints in a straightforward manner by means of a Lagrange-multiplier technique. Coils of genuinely finite length are accounted for without approximation by this technique, and it therefore has distinct advantages over the target field method (and alternative methods based on finite-elements). Since it relies on a stochastic optimization strategy, it can even cope with discontinuous objective functions, and so can accommodate adding or removing loops of wire during the optimization process. The method has the drawback that the stochastic optimization technique can take many iterations to converge, and so can be expensive of computer time. In addition, the technique is undoubtedly more difficult to apply to the design of coils that produce more complicated magnetic fields, such as those involved in higher-order spherical harmonics with tesseral components.
It is an object of the invention to provide coil structures that generate desired fields internal or external to the coil structure, that may be symmetric or non-symmetric with respect to that structure. For example, in connection with certain preferred embodiments, it is an object of the invention to provide coil structures that generate desired fields within certain specific and asymmetric portions of the coil structure.
It is a further object of the invention to provide a general systematic method for producing any desired zonal or tesseral or otherwise shaped magnetic field within and/or outside a coil, taking the finite length of the coil into account without approximation.
In one broad form, the invention provides a method for the design of coils for the production of magnetic fields. For example, such coils can be shim coils of the type suitable for use in Magnetic Resonance applications. The method involves a type of target-field approach, but the exact geometry of the coils is treated without approximation. In particular, the fact that coils are of finite length is catered for.
Target fields of any desired type may be specified, and may involve zonal and tesseral harmonics or any other specified field shape. The method of this invention can be used to design the coil windings needed to produce the specified target field. In this approach, there is complete freedom in the choice of target field. For example, there is no requirement to restrict the target field to any one spherical harmonic. The method is therefore able to design coils in which the region of interest is located asymmetrically with respect to the coil length. In addition, to improve the accuracy of the fields produced by the coil, the design methodology of this invention can match desired target fields at two or more different target radii, which preferably are co-axial.
In one embodiment, the invention provides a method for designing a coil, e.g., a tesseral shim coil for a magnetic resonance system, where the coil extends from xe2x88x92L to +L along a longitudinal axis which lies along the z-axis of a three dimensional coordinate system, and the method comprises the steps of:
(a) selecting a cylindrical surface having a radius r=a for calculating current densities for the coil (the xe2x80x9cr=a surfacexe2x80x9d), the surface surrounding the longitudinal axis and extending from xe2x88x92L to +L;
(b) selecting a set of desired values for the longitudinal component of the magnetic field Bz (or HT)to be produced by the coil at locations which are spaced along the longitudinal axis from z=pL to z=qL where xe2x88x921 less than p less than q less than 1 (for example, the desired values for the longitudinal component of the magnetic field can be defined by a preselected single tesseral or combinations of tesseral harmonics); and
(c) determining a current density distribution j(xcex8,z) for the coil by:
(1) establishing equations for the relationships between the current density and the target fields (see, for example, equations 4.9-4.12 herein); and
(2) solving said equations using a matrix regularization method (see, for example, equations 4.13-4.17), wherein the regularized expression to be minimized, in one preferred embodiment, is the curvature of a streamfunction defined by, for example, equations 4.18 and 4.19 set forth below.
In other embodiments, the quantities for minimization in the regularization procedure can be the power and/or energy contained in the device (see, for example, equation 4.14).
The procedures outlined in (1) and (2) above can be preferably used for multiple target field regions (see, for example, equations 4.20-4.23)
The method preferably also includes the additional step of generating discrete current carrying windings for the coil from the current density distribution j(xcex8,z) by:
(1) using the current density vector j(xcex8,z) to create a streamfunction "psgr" according to, for example, equations 4.16 and 4.17
(2) selecting a number of current carrying windings N;
(3) determining a current per winding value I=J/N, where J is the total obtained by integrating the current density vector over the r=a surface (the xe2x80x9ctotal integrated currentxe2x80x9d);
(4) contouring the streamfunction * and thereby determining a set of j(xcex8,z) blocks over the r=a surface (i.e., the surface of the current density cylinder) within the longitudinal range from xe2x88x92L to +L such that the integral of j(xcex8,z) over each block equals I; and
(5) for all blocks having a net polarity for j(xcex8,z) over the block, placing a winding at the center of the block, the direction of the current in the winding corresponding to said net polarity.
The method can be used for symmetrical and asymmetrical cases, i.e., |p|=|q| and |p|xe2x89xa0|q|, respectively.
In one of its general method aspects, the invention provides a method for designing a coil, where the coil has a longitudinal axis which lies along the z-axis of a three dimensional coordinate system, said method comprising the steps of:
(a) selecting one or more cylindrical surfaces Si (ixe2x89xa71) for calculating current densities for the coil, each surface (1) having a radius ai, (2) surrounding the longitudinal axis, and (3) extending from xe2x88x92Li to +Li;
(b) selecting a set of desired values for the longitudinal component of the magnetic field Bz (or HT) to be produced by the coil at target locations on one or more cylindrical surfaces Tj (jxe2x89xa71) which extend along the longitudinal axis from z=pLi to z=qLi where i=1 and xe2x88x921 less than p less than q less than 1; and
(c) determining current density distributions ji(xcex8,z) at the Si surfaces for the coil by:
(1) expressing Bz (or HT) in terms of ji(xcex8,z) (see, for example, equations 4.1 to 4.4 and 4.6);
(2) expressing the z-component of each of the ji(xcex8,z) (jzi(xcex8,z)) in terms of a basis function expansion and constraining jzi(xcex8,z) to equal 0 at xe2x88x92Li and +Li, said expansion including a set of coefficients (see, for example, equation 4.7);
(3) expressing Bz (or HT) in terms of said basis function expansions (see, for example, equations 4.5, 4.8, 4.9 and 4.10); and
(4) for each of jzi(xcex8,z), determining values for said set of coefficients of said basis function expansion by:
i. selecting an error function which comprises a measure of the difference between the desired Bz (or HT) values at the target locations and the Bz (or HT) values calculated at those locations using the basis function expansions of the jzi(xcex8,z) (see, for example, equations 4.11 and 4.12);
ii. regularizing said error function (see, for example, equation 4.13); and
iii. determining values for the sets of coefficients using the regularized error function (see, for example, equation 4.14-4.15); and
(5) determining the azimuthal-component of ji(xcex8,z) (jxcex8i(xcex8,z)) using the values for the sets of coefficients determined in step (c)(4) and the continuity equation.
In accordance with another of its general method aspects, the invention provides a method for designing a coil comprising:
(a) selecting one or more surfaces Si (ixe2x89xa71) for calculating current densities for the coil;
(b) selecting a set of desired values for one or more components of the magnetic field B (or H) to be produced by the coil at target locations on one or more surfaces Tj(jxe2x89xa71); and
(c) determining current density distributions jiat the Si surfaces for the coil by:
(1) expressing B (or H) in terms of ji;
(2) expressing each of the ji in terms of a basis function expansion having a set of coefficients;
(3) expressing B (or H) in terms of said basis function expansions; and
(4) for each ji determining values for said set of coefficients of said basis function expansion by:
i. selecting an error function which comprises a measure of the difference between the desired B (or H) values at the target locations and B (or H) values calculated at those locations using the basis function expansions of the ji;
ii. regularizing said error function using one or more regularization parameters, one parameter for each ji, said parameter being the curvature of a streamfunction for ji(see, for example, equations 4.16-4.19); and
iii. determining values for the sets of coefficients using the regularized error function.
The various preferred and other embodiments discussed above and below apply to these general forms of the method aspects of the invention.
In another broad form, the invention provides coils, e.g., shim coils, for the production of tesseral magnetic fields located asymmetrically in a finite-length coil. As is well known in the art, a zonal field has complete azimuthal symmetry, i.e., it is not a function of xcex8 in a conventional cylindrical coordinate system, while a tesseral field does not have complete azimuthal symmetry, i.e., it depends on xcex8.
In one embodiment, the invention provides a tesseral coil (e.g., a member of a shim set) having (i) a longitudinal axis (e.g., the z-axis) and a radius r describing a primary cylindrical surface and (ii) a predetermined volume in which at least one predetermined tesseral harmonic is generated (the tesseral harmonic volume; e.g., a shimming volume), and comprising a plurality of current-carrying windings (e.g., interconnected, arc-like, windings) associated with the primary cylindrical surface (e.g., mounted on and/or mounted in and/or mounted to a support member located at the cylindrical surface), the tesseral coil producing a magnetic field, the longitudinal component of which is given by:                               B          ⁡                      (                          r              ,              θ              ,              φ                        )                          =                              ∑                          n              =              0                        ∞                    ⁢                                    ∑                              m                =                0                                            m                =                n                                      ⁢                                                            r                  n                                ⁡                                  [                                                                                    A                                                  n                          ⁢                                                      xe2x80x83                                                    ⁢                          m                                                                    ⁢                                              cos                        ⁡                                                  (                                                      m                            ⁢                                                          xe2x80x83                                                        ⁢                            φ                                                    )                                                                                      +                                                                  B                                                  n                          ⁢                                                      xe2x80x83                                                    ⁢                          m                                                                    ⁢                                              sin                        ⁡                                                  (                                                      m                            ⁢                                                          xe2x80x83                                                        ⁢                            φ                                                    )                                                                                                      ]                                            ⁢                                                P                                      n                    ⁢                                          xe2x80x83                                        ⁢                    m                                                  ⁡                                  (                                      cos                    ⁢                                          xe2x80x83                                        ⁢                    θ                                    )                                                                                        (        1        )            
where Anm and Bnm are the amplitudes of spherical harmonics, Pnm(cos xcex8) are associated Legendre polynomials, n is the order and m the degree of the polynomial, and r ,xcex8 and xcfx86 are polar (spherical) co-ordinates;
and wherein:
(i) the at least one predetermined tesseral harmonic has a degree mxe2x80x2 and an order nxe2x80x2 which satisfy the relationships:
mxe2x80x2 greater than 0, and
nxe2x80x2xe2x89xa72;
(ii) the primary cylindrical surface has first and second ends which define a length 2L therebetween; and
(iii) the tesseral harmonic volume extends along the longitudinal axis from z=pL to z=qL, where
(a) xe2x88x921 less than p less than q less than 1;
(b) |p|xe2x89xa0|q| (i.e., the tesseral harmonic volume is located asymmetrically with respect to the overall geometry of the coil); and
(c) z=0 is midway between the first and second ends of the primary cylindrical surface.
In certain embodiments of the invention, the tesseral coil generates at least one additional predetermined tesseral harmonic in the tesseral harmonic volume, said at least one additional harmonic having a degree different from mxe2x80x2 and/or an order different from nxe2x80x2
Preferably, the tesseral coil is a shim coil and most preferably, the shim coil is a member of a shim set for which all of the tesseral coils in the set are of the above type.
In the case of tesseral shim coils used for high resolution spectroscopy, i.e., NMR, qxe2x88x92p is preferably greater than or equal to 0.01 and most preferably greater than or equal to 0.05. In the case of tesseral shim coils used for clinical imaging, i.e., MRI, qxe2x88x92p is preferably greater than or equal to 0.05 and most preferably greater than or equal to 0.5.
In accordance with certain preferred embodiments of the invention, the tesseral coil generates a single predetermined tesseral harmonic, the tesseral harmonic volume defines a midpoint M along the longitudinal axis, the volume has a characteristic radius c given by:
c=(qxe2x88x92p)L/2 when qxe2x88x92p less than 1,xe2x80x83xe2x80x83(2)
and by:
c=(qxe2x88x92p)L/3 when qxe2x88x92pxe2x89xa71,xe2x80x83xe2x80x83(3)
and the tesseral coil has a purity (Pxe2x80x2) which is less than or equal to 0.2, where Pxe2x80x2 equals the ratio of (1) the sum of the magnitudes of all harmonic coefficients other than the coefficient of the predetermined tesseral harmonic which have a magnitude which is at least 0.001% of the magnitude of the coefficient of the predetermined tesseral harmonic to (2) the magnitude of the coefficient of the predetermined tesseral harmonic..
Most preferably, Pxe2x80x2 is less than or equal to 0.05.
In certain specific applications of the invention, the tesseral coil has the following characteristics:
(i) nxe2x80x2=2or 3;
(ii) qxe2x88x92pxe2x89xa70.7;
(iii) 2Lxe2x89xa61.4 meters; and
(iv) Pxe2x80x2xe2x89xa60.1;
while in other specific applications, it has the following characteristics:
(i) nxe2x80x2=4, 5, 6, 7, or 8;
(ii) qxe2x88x92pxe2x89xa70.7;
(iii) 2Lxe2x89xa61.4 meters; and
(iv) Pxe2x80x2xe2x89xa60.2.
In each case, mxe2x80x2 will typically be less than or equal to nxe2x80x2.
In other preferred embodiments, the tesseral coil produces a plurality of predetermined tesseral harmonics, the tesseral harmonic volume defines a midpoint M along the longitudinal axis, the tesseral harmonic volume has a characteristic radius c given by:
c=(qxe2x88x92p)L/2 when qxe2x88x92p less than 1,
and by:
c=(qxe2x88x92p)L/3 when qxe2x88x92pxe2x89xa71;
and the tesseral coil has a purity (Pxe2x80x2) which is less than or equal to 0.2 (preferably less than or equal to 0.05), where Pxe2x80x2 equals the ratio of (1) the sum of the magnitudes of all harmonic coefficients other than the coefficients of the plurality of predetermined tesseral harmonics which have a magnitude which is at least 0.001% of the magnitude of the largest coefficient of the plurality of predetermined tesseral harmonics to (2) the sum of the magnitudes of the coefficients of the plurality of predetermined tesseral harmonics.
In still further preferred embodiments, the tesseral coil further comprises a shielding cylindrical surface co-axial and external to the primary cylindrical surface, said shielding cylindrical surface having a plurality of current-carrying windings associated therewith, said windings of the primary and shielding cylindrical surfaces causing the magnitude of the magnetic field generated by the tesseral coil to be below a predetermined value (preferably effectively zero) outside of a predetermined surface external to the shielding cylindrical surface. In connection with these embodiments, the shielding cylindrical surface has first and second ends which define a length 2Lxe2x80x2 therebetween, where Lxe2x80x2=rbL and rb is preferably greater than or equal to 1.0.
For clinical imaging applications of the invention, either |p| or |q| is preferably greater than or equal to 0.7.
Asymmetric tesseral shim coils may be used in compact conventional magnet systems such as those of U.S. Pat. No. 5,818,319 or alternately, they may be used in asymmetric magnets, such as the magnets of U.S. Pat. No. 6,140,900.